We prove interpolation inequalities by means of the Lorentz norm, BMO norm, and the fractional Sobolev norm

Electronic Journal of Differential Equations, Vol. 2019 (2019), No. 56, pp. 1–4.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

INTERPOLATION INEQUALITIES BETWEEN LORENTZ SPACE AND BMO: THE ENDPOINT CASE (L^1,∞, BM O)

NGUYEN ANH DAO, NGUYEN THI NGOC HANH, TRAN MINH HIEU, HUY BAC NGUYEN

Communicated by Jesus Ildefonso Diaz

Abstract. We prove interpolation inequalities by means of the Lorentz norm, BMO norm, and the fractional Sobolev norm. In particular, we obtain an interpolation inequality for (L^1,∞, BM O), that we call the endpoint case.

1. Introduction and statement of main results

The main purpose of this article is to study the interpolation inequalities between the Lorentz space L^p,α(Rⁿ) and the BM O(Rⁿ) space, where n ≥1. It is known that the interpolation inequalities play a crucial role in studying the boundedness of operators and in studying PDEs, see, e.g. [1, 2, 5, 6, 7, 8]. Thus, such an extension of the inequalities of this type is involved many purposes, for instance: the theory of Marcinkiewicz interpolation; the boundedness of the operators acting on Lorentz spaces (the Hardy-Littlewood maximal function, the Hilbert transform, and the Riesz transform); and the estimates in PDEs.

In this article, we want to prove an interpolation inequality between the Lorentz space L^q,α(Rⁿ) andBM O(Rⁿ), forq ≥1, and α >0. And we call the endpoint case whenq= 1. Our result is as follows.

Theorem 1.1. Let 1 ≤q < p, and 0< α <∞. Let f ∈L^q,∞(Rⁿ)∩BM O(Rⁿ).

Then

kfkL^p,α(Rⁿ).kfk^q/p_Lq,∞(Rⁿ)kfk¹⁻

q p

BM O(Rⁿ). (1.1)

This result extends the recent results in [2, 3]. As a consequence of Theorem 1.1, we obtain an interpolation inequality betweenL^q,∞ and the critical Sobolev space W˙ ^s,ns(Rⁿ) fors∈(0,1).

Corollary 1.2. Let 1≤q < p, andα >0. For any0< s <1, we have kfk_Lp,α(Rⁿ).kfk^q/p_Lq,∞(Rⁿ)kfk¹⁻

q p

W˙^{s, n}s(Rⁿ). (1.2)

2010Mathematics Subject Classification. 46E35, 26D10.

Key words and phrases. Gagliardo-Nirenberg inequality; Lorentz spaces; BMO space;

Fractional Sobolev spaces.

c

2019 Texas State University.

Submitted February 6, 2019. Published May 3, 2019.

1

2 N. A. DAO, N. T. N. HANH, T. M. HIEU, H. B. NGUYEN EJDE-2019/56

Note that (1.2) follows from (1.1) and the inclusion ˙W^s,ns(Rⁿ)⊂BM O(Rⁿ).

Before proving Theorem 1.1, we recall the definitions of the Lorentz spaces, and BM Ospace. Givenq, α >0, we set

kgk_Lq,α(Rⁿ):=





 qR∞

0 (λ^q|{x∈Rⁿ:|g(x)|> λ}|)^{α/q dλ}_λ1/α

ifα <∞, sup_λ>0λ |{x∈Rⁿ :|g(x)|> λ}|^1/q

ifα=∞.

The Lorentz space is L^q,α(Rⁿ) = {g : Rⁿ → R: kgk_Lq,α(Rⁿ) < ∞}. Next, we define the sharp maximal function:

f^](x) = sup

R>0,x∈BR

1

|B_R| Z

BR

|f(y)−(f)B_R|dy, with (f)_Ω= _|Ω|¹ R

Ωf(x)dx. Then, we have a result, the so called strong type (p, p) inL^p(Rⁿ) as follows (see, e.g. [9]).

Theorem 1.3. Let p >1. Then

kfk_Lp(Rⁿ).kf^]k_Lp(Rⁿ), (1.3) whenever the right hand side is well-defined.

After that, we denote by BM O(Rⁿ) =

f ∈L¹_loc(Rⁿ) :kfkBM O(Rⁿ)= sup

x∈Rⁿ

f^](x)<∞ . Finally, we denote the homogeneous fractional Sobolev space by

W˙ ^s,p(Rⁿ)

=

f ∈ S⁰(Rⁿ) :kfkW˙^s,p(Rⁿ)=Z

Rⁿ

Z

Rⁿ

|f(x)−f(y)|^p

|x−y|^n+sp dx dy1/p

<∞ , where S⁰(Rⁿ) is the dual space of S(Rⁿ) (the Schwartz space). To end this part, we denoteA.B ifA≤cB, wherec >0 is a constant.

2. Proof of Theorem 1.1

It suffices to show that (1.1) holds for q = 1. To start, we prove the following result.

Lemma 2.1. Let 0 < q < p < r ≤ ∞and α >0. Iff ∈L^q,∞(Rⁿ)∩L^r,∞(Rⁿ), thenf ∈L^p,α(Rⁿ), and

kfkL^p,α(Rⁿ).kfk^θ_Lq,∞(Rⁿ)kfk^1−θ_Lr,∞(Rⁿ), (2.1) with ¹_p = ^θ_q +^1−θ_r .

Proof. We rewrite kfk^α_Lp,α(Rⁿ)=p

Z λ₀ 0

λ^α|{|f|> λ}|^α/pdλ λ +p

Z ∞ λ0

λ^α|{|f|> λ}|^α/pdλ

λ. (2.2) Sincef ∈L^q,∞(Rⁿ)∩L^r,∞(Rⁿ), we have

Z λ₀ 0

λ^α|{|f|> λ}|^α/pdλ

λ ≤

Z λ₀ 0

λ^αkfk^q_Lq,∞(Rⁿ)

λ^q

^α/pdλ λ

=kfk^αq/p_Lq,∞(Rⁿ)

α(1−q/p) λ^α(1−q/p)₀ ,

(2.3)

EJDE-2019/56 INTERPOLATION INEQUALITIES 3

and

Z ∞ λ₀

λ^α|{|f|> λ}|^α/pdλ

λ ≤

Z ∞ λ₀

λ^αkfk^r_Lr,∞(Rⁿ)

λ^r

^α/pdλ λ

=

kfk^αr/p_Lr,∞(Rⁿ)

α(r/p−1)λ^α(1−r/p)₀ .

(2.4)

By (2.2), (2.3) and (2.4), we obtain

kfk^α_Lp,α(Rⁿ)≤pkfk^αq/p_Lq,∞(Rⁿ)

α(1−q/p)λ^α(1−q/p)₀ +

kfk^αr/p_Lr,∞(Rⁿ)

α(r/p−1)λ^α(1−r/p)₀ .

Now, we take

λ^r−q₀ = kfk^r_Lr,∞(Rⁿ)

kfk^q_Lq,∞(Rⁿ)

,

so the proof is complete.

Thanks to Lemma 2.1, we have for anyr > p

kfk_Lp,α(Rⁿ).kfk^θ_L1,∞(Rⁿ)kfk^1−θ_Lr,∞(Rⁿ), (2.5) where ¹_p =θ+^1−θ_r .

Sincer > p >1, and by (1.3), we obtain

kfk^r_Lr,∞(Rⁿ)≤ kfk^r_Lr(Rⁿ).kf^]k^r_Lr(Rⁿ)

.kfk^r−p_{BM O(}

Rⁿ)kf^]k^p_Lp(Rⁿ)

.kfk^r−p_{BM O(}

Rⁿ)kfk^p_Lp(Rⁿ).

(2.6)

Combining (2.5) and (2.6) yields kfk_Lp,α(Rⁿ).kfk^θ_L1,∞(Rⁿ)

kfk¹⁻

p r

BM O(Rⁿ)kfk

p r

L^p(Rⁿ)

1−θ

.kfk^θ_L1,∞(Rⁿ)

kfk¹⁻_{BM O(}^pr

Rⁿ)kfk_L^pr_p,α(

Rⁿ)

1−θ

. Then

kfk¹⁻_Lp,α^pr(1−θ)(Rⁿ) .kfk^θ_L1,∞(Rⁿ)kfk⁽¹⁻_{BM O(}^pr)(1−θ)

Rⁿ) , kfk_Lp,α(Rⁿ).kfk

1 p

L^1,∞(Rⁿ)kfk¹⁻

1 p

BM O(Rⁿ). Thus, the proof is complete.

References

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[2] D. S. Mc Cormick, J. C. Robinson, J. L. Rodrigo; Generalised Gagliardo -Nirenberg In- equalities Using Weak Lebesgue Spaces and BMO, Milan Journal of Mathematics,81(2013), 265-289.

[3] Nguyen Anh Dao, Jesus Ildefonso D´ıaz, Quoc-Hung Nguyen;Generalized Gagliardo-Nirenberg inequalities using Lorentz spaces,BM O, H¨older spaces and fractional Sobolev spaces, Non- linear Analysis,173(2018), 146-153.

[4] F. John, L. Nirenberg;On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14(1961), 415-426.

[5] H. Kozono, Y. Taniuchi;Limiting case of the Sobolev inequality inBM Owith application to the Euler equations, Commun. Math. Phys.,214(2000), 191-200.

4 N. A. DAO, N. T. N. HANH, T. M. HIEU, H. B. NGUYEN EJDE-2019/56

[6] H. Kozono, H. Wadade;Remarks on Gagliardo-Nirenberg type inequality with critical Sobolev space and BMO, Math. Zeit.,295(2008), 935-950.

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[8] T. Ogawa, T. Ozawa;Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schr¨odinger mixed problem, J. Math. Anal. Appl.,155(1991), 531-540.

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Nguyen Anh Dao

Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

Email address:[email protected]

Nguyen Thi Ngoc Hanh

Le Loi High school, Gia Lai Province, Vietnam

Email address:[email protected]

Tran Minh Hieu

Luong The Vinh High school, Gia Lai Province. Vietnam Email address:[email protected]

Huy Bac Nguyen

Faculty of Electrical Engineering & Computer Science, Technical University of Os- trava, Czech Republic

Email address:[email protected]